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annotate OD.agda @ 290:359402cc6c3d

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definition of filter
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 12 Jun 2020 19:19:16 +0900
parents d9d3654baee1
children ef93c56ad311
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
2 open import Ordinals
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
3 module OD {n : Level } (O : Ordinals {n} ) where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
5 open import zf
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
7 open import Relation.Binary.PropositionalEquality
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
8 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
9 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
10 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
11 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
14 open import logic
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
15 open import nat
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
16
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
17 open inOrdinal O
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
18
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
19 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
20
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
21 record OD : Set (suc n ) where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
22 field
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
23 def : (x : Ordinal ) → Set n
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
24
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
25 open OD
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
26
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
27 open _∧_
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
28 open _∨_
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
29 open Bool
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fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
30
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
31 record _==_ ( a b : OD ) : Set n where
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
32 field
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
33 eq→ : ∀ { x : Ordinal } → def a x → def b x
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
34 eq← : ∀ { x : Ordinal } → def b x → def a x
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
35
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
36 id : {A : Set n} → A → A
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
37 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
38
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
39 ==-refl : { x : OD } → x == x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
40 ==-refl {x} = record { eq→ = id ; eq← = id }
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
41
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
42 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
43
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
44 ==-trans : { x y z : OD } → x == y → y == z → x == z
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
45 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) }
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
46
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
47 ==-sym : { x y : OD } → x == y → y == x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
48 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
49
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
50
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
51 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
52 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
53 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
54
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
55 -- next assumptions are our axiom
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
56 -- In classical Set Theory, HOD is used, as a subset of OD,
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
57 -- HOD = { x | TC x ⊆ OD }
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
58 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
59 -- This is not possible because we don't have V yet.
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
60 -- We simply assume V=OD here.
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
61 --
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
62 -- We also assumes ODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
63 -- ODs have an ovbious maximum, but Ordinals are not. This means, od→ord is not an on-to mapping.
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
64 --
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
65 -- ==→o≡ is necessary to prove axiom of extensionality.
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
66 --
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
67 -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic,
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
68 -- we need explict assumption on sup.
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
69
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
70 record ODAxiom : Set (suc n) where
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
71 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
72 field
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
73 od→ord : OD → Ordinal
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
74 ord→od : Ordinal → OD
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
75 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
76 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
77 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
78 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
79 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
80 sup-o : ( OD → Ordinal ) → Ordinal
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
81 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
82 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
83 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
84 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
85
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
86 postulate odAxiom : ODAxiom
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
87 open ODAxiom odAxiom
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
88
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
89 data One : Set n where
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
90 OneObj : One
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
91
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
92 -- Ordinals in OD , the maximum
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
93 Ords : OD
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
94 Ords = record { def = λ x → One }
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
95
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
96 maxod : {x : OD} → od→ord x o< od→ord Ords
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
97 maxod {x} = c<→o< OneObj
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
98
179
aa89d1b8ce96 fix comments
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
99 -- Ordinal in OD ( and ZFSet ) Transitive Set
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
100 Ord : ( a : Ordinal ) → OD
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
101 Ord a = record { def = λ y → y o< a }
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
102
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
103 od∅ : OD
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
104 od∅ = Ord o∅
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
105
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
106
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
107 o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
108 o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
109 lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
110 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
111 lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
112 lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt )
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
113
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
114 _∋_ : ( a x : OD ) → Set n
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
115 _∋_ a x = def a ( od→ord x )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
116
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
117 _c<_ : ( x a : OD ) → Set n
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
118 x c< a = a ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
119
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
120 cseq : {n : Level} → OD → OD
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
121 cseq x = record { def = λ y → def x (osuc y) } where
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
122
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
123 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
124 def-subst df refl refl = df
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
125
260
8b85949bde00 sup with limit give up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 259
diff changeset
126 sup-od : ( OD → OD ) → OD
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
127 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
128
260
8b85949bde00 sup with limit give up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 259
diff changeset
129 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
130 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
131 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
132 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x))
260
8b85949bde00 sup with limit give up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 259
diff changeset
133 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) )
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
134
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
135 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y
187
ac872f6b8692 add Todo
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
136 otrans x<a y<x = ordtrans y<x x<a
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
137
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
138 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
139 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
140
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
141
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
142 -- avoiding lv != Zero error
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
143 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
144 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
145
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
146 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
147 ==-iso {x} {y} eq = record {
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
148 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
149 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
150 where
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
151 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
152 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
153
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
154 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y)
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
155 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso)
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
156
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
157 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
158 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
159 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
160 lemma ox ox refl = ==-refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
161
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
162 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
163 o≡→== {x} {.x} refl = ==-refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
164
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
165 o∅≡od∅ : ord→od (o∅ ) ≡ od∅
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
166 o∅≡od∅ = ==→o≡ lemma where
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
167 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
168 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
169 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
170 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
171 lemma : ord→od o∅ == od∅
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
172 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
173
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
174 ord-od∅ : od→ord (od∅ ) ≡ o∅
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
175 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
176
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
177 ∅0 : record { def = λ x → Lift n ⊥ } == od∅
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
178 eq→ ∅0 {w} (lift ())
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
179 eq← ∅0 {w} lt = lift (¬x<0 lt)
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
180
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
181 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ )
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
182 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
183 ∅< {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
184
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
185 ∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
186 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
187
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
188 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
189 def-iso refl t = t
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
190
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
191 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
192 is-o∅ x with trio< x o∅
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
193 is-o∅ x | tri< a ¬b ¬c = no ¬b
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
194 is-o∅ x | tri≈ ¬a b ¬c = yes b
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
195 is-o∅ x | tri> ¬a ¬b c = no ¬b
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
196
254
2ea2a19f9cd6 ordered pair clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 247
diff changeset
197 _,_ : OD → OD → OD
2ea2a19f9cd6 ordered pair clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 247
diff changeset
198 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y))
188
1f2c8b094908 axiom of choice → p ∨ ¬ p
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
199
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
200 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
201 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
59
d13d1351a1fa lemma = cong₂ (λ x not → minimul x not ) oiso { }6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 58
diff changeset
202
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
203 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
204 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
205
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
206 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
207
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
208 ZFSubset : (A x : OD ) → OD
191
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
209 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
210
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
211 Def : (A : OD ) → OD
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
212 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) )
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
213
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
214 -- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
215 -- _⊆_ A B {x} = A ∋ x → B ∋ x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
216
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
217 record _⊆_ ( A B : OD ) : Set (suc n) where
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
218 field
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
219 incl : { x : OD } → A ∋ x → B ∋ x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
220
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
221 open _⊆_
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
222
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
223 infixr 220 _⊆_
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
224
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
225 subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
226 subset-lemma {A} {x} = record {
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
227 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
228 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
229 }
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
230
261
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
231 open import Data.Unit
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
232
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
233 ε-induction : { ψ : OD → Set (suc n)}
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
234 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
235 → (x : OD ) → ψ x
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
236 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
237 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
238 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
239 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
240 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
241
262
53744836020b CH trying ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 261
diff changeset
242 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
53744836020b CH trying ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 261
diff changeset
243 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} )
53744836020b CH trying ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 261
diff changeset
244
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
245 OD→ZF : ZF
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
246 OD→ZF = record {
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
247 ZFSet = OD
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
248 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
249 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
250 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
251 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
252 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
253 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
254 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
255 ; Replace = Replace
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
256 ; infinite = infinite
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
257 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
258 } where
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
259 ZFSet = OD
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
260 Select : (X : OD ) → ((x : OD ) → Set n ) → OD
156
3e7475fb28db differeent Union approach
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 155
diff changeset
261 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
262 Replace : OD → (OD → OD ) → OD
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
263 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x }
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
264 _∩_ : ( A B : ZFSet ) → ZFSet
145
f0fa9a755513 all done but ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
265 A ∩ B = record { def = λ x → def A x ∧ def B x }
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
266 Union : OD → OD
156
3e7475fb28db differeent Union approach
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 155
diff changeset
267 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
268 _∈_ : ( A B : ZFSet ) → Set n
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
269 A ∈ B = B ∋ A
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
270 Power : OD → OD
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
271 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
272 -- ｛_｝ : ZFSet → ZFSet
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
273 -- ｛ x ｝ = ( x , x )
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
274
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
275 data infinite-d : ( x : Ordinal ) → Set n where
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
276 iφ : infinite-d o∅
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
277 isuc : {x : Ordinal } → infinite-d x →
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
278 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
279
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
280 infinite : OD
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
281 infinite = record { def = λ x → infinite-d x }
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
282
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
283 infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
284 -- infixr 230 _∩_ _∪_
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
285 isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
286 isZF = record {
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
287 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
247
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
288 ; pair→ = pair→
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
289 ; pair← = pair←
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
290 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
291 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
292 ; empty = empty
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
293 ; power→ = power→
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
294 ; power← = power←
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
295 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
274
29a85a427ed2 ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
296 ; ε-induction = ε-induction
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
297 ; infinity∅ = infinity∅
160
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 159
diff changeset
298 ; infinity = infinity
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
299 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
135
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
300 ; replacement← = replacement←
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
301 ; replacement→ = replacement→
274
29a85a427ed2 ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
302 -- ; choice-func = choice-func
29a85a427ed2 ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
303 -- ; choice = choice
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
304 } where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
305
247
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
306 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
307 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x ))
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
308 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y ))
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
309
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
310 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
311 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x))
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
312 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y))
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
313
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
314 empty : (x : OD ) → ¬ (od∅ ∋ x)
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
315 empty x = ¬x<0
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
316
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
317 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
318 o<→c< lt = record { incl = λ z → ordtrans z lt }
155
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
319
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
320 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
155
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
321 ⊆→o< {x} {y} lt with trio< x y
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
322 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
323 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
324 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
155
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
325 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
151
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
326
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
327 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
157
afc030b7c8d0 explict logical definition of Union failed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 156
diff changeset
328 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
afc030b7c8d0 explict logical definition of Union failed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 156
diff changeset
329 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
159
3675bd617ac8 infinite continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 158
diff changeset
330 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
331 union← X z UX∋z = FExists _ lemma UX∋z where
165
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 162
diff changeset
332 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 162
diff changeset
333 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
334
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
335 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
336 ψiso {ψ} t refl = t
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
337 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
338 selection {ψ} {X} {y} = record {
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
339 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
340 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
341 }
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
342 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x
260
8b85949bde00 sup with limit give up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 259
diff changeset
343 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
344 lemma : def (in-codomain X ψ) (od→ord (ψ x))
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
345 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
346 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
347 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
348 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
349 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
350 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
351 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y))
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
352 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq )
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
353 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
354 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 ))
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
355
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
356 ---
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
357 --- Power Set
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
358 ---
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
359 --- First consider ordinals in OD
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
360 ---
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
361 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
362 --
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
363 --
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
364 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
365 ∩-≡ {a} {b} inc = record {
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
366 eq→ = λ {x} x<a → record { proj2 = x<a ;
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
367 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
368 eq← = λ {x} x<a∩b → proj2 x<a∩b }
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
369 --
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
370 -- Transitive Set case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
371 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
372 -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
373 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
374 --
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
375 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
376 ord-power← a t t→A = def-subst {_} {_} {Def (Ord a)} {od→ord t}
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
377 lemma refl (lemma1 lemma-eq )where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
378 lemma-eq : ZFSubset (Ord a) t == t
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
379 eq→ lemma-eq {z} w = proj2 w
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
380 eq← lemma-eq {z} w = record { proj2 = w ;
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
381 proj1 = def-subst {_} {_} {(Ord a)} {z}
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
382 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
383 lemma1 : {a : Ordinal } { t : OD }
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
384 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
385 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
386 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x))
260
8b85949bde00 sup with limit give up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 259
diff changeset
387 lemma = sup-o<
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
388
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
389 --
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
390 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
391 -- then replace of all elements of the Power set by A ∩ y
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
392 --
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
393 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y )
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
394
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
395 -- we have oly double negation form because of the replacement axiom
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
396 --
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
397 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
398 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
399 a = od→ord A
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
400 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y)))
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
401 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
402 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
403 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
404 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
405 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
406 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x))
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
407 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
408
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
409 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
410 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
411 a = od→ord A
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
412 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
413 lemma0 {x} t∋x = c<→o< (t→A t∋x)
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
414 lemma3 : Def (Ord a) ∋ t
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
415 lemma3 = ord-power← a t lemma0
152
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
416 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
417 lemma4 = let open ≡-Reasoning in begin
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
418 A ∩ ord→od (od→ord t)
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
419 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
420 A ∩ t
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
421 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
422 t
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
423 ∎
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
424 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x))
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
425 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x)))
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
426 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} )
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
427 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t)
151
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
428 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
429 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
430 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A )))
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
431
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
432 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
433 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
434 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
435 lemma lt y<x with osuc-≡< lt
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
436 lemma lt y<x | case1 refl = c<→o< y<x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
437 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
262
53744836020b CH trying ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 261
diff changeset
438
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
439 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
440 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
441
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
442 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
443 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
444 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
445
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
446 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
447 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
448 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
449
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
450 infinity∅ : infinite ∋ od∅
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
451 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
452 lemma : o∅ ≡ od→ord od∅
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
453 lemma = let open ≡-Reasoning in begin
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
454 o∅
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
455 ≡⟨ sym diso ⟩
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
456 od→ord ( ord→od o∅ )
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
457 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
458 od→ord od∅
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
459 ∎
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
460 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
461 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
462 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
463 ≡ od→ord (Union (x , (x , x)))
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
464 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
465
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
466
226
176ff97547b4 set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 223
diff changeset
467 Union = ZF.Union OD→ZF
176ff97547b4 set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 223
diff changeset
468 Power = ZF.Power OD→ZF
176ff97547b4 set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 223
diff changeset
469 Select = ZF.Select OD→ZF
176ff97547b4 set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 223
diff changeset
470 Replace = ZF.Replace OD→ZF
176ff97547b4 set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 223
diff changeset
471 isZF = ZF.isZF OD→ZF